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We also prove that there exists a positive density of $n$ such that $P^+(n)<P^+(n+1)<x^{41/107+\\varepsilon}$. Define $T_c(x):=\\#\\{p\\leq x:P^+(p-1)\\geq p^c\\}$. For $1/2<c<1$, we also show that \\begin{align*}\n  \\mathop{\\lim \\sup}_{x\\rightarrow\\infty}\\frac{T_c(x)}{\\pi(x)}\\leq \\min\\left("},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.16032","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-17T15:07:16Z","cross_cats_sorted":[],"title_canon_sha256":"88d992ff4f5cf7603a2dda16be29d90d8629f051a637410925564d8a544408cc","abstract_canon_sha256":"a516fb3c9aafdb35233c26c7744db85ce891a6a638ab6be9f6584540aa6d0db8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-20T02:19:27.350031Z","signature_b64":"NW2DMmcrCPdwdmgKxBWPOzUkEPNou7C9N8cVYigmm3rVBijxGCuIrcyNQwLcABj7W8yzWvJLohLUEnYHh9cGAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3454472e5d69e88fbcab2d18f976f93025688b7f05ca4684341621e9b166e111","last_reissued_at":"2026-07-20T02:19:27.349169Z","signature_status":"signed_v1","first_computed_at":"2026-07-20T02:19:27.349169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An improvement on the largest prime factors of consecutive integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhiyuan Yang","submitted_at":"2026-07-17T15:07:16Z","abstract_excerpt":"Let $P^+(n)$ denote the largest prime factor of $n$. One of Erd\\H{o}s and Tur\\'an's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is 1/2. In this paper, we prove that this density is larger than 0.280, which improves the previous result 0.2017 by L\\\"u and Wang (2025). We also prove that there exists a positive density of $n$ such that $P^+(n)<P^+(n+1)<x^{41/107+\\varepsilon}$. Define $T_c(x):=\\#\\{p\\leq x:P^+(p-1)\\geq p^c\\}$. For $1/2<c<1$, we also show that \\begin{align*}\n  \\mathop{\\lim \\sup}_{x\\rightarrow\\infty}\\frac{T_c(x)}{\\pi(x)}\\leq \\min\\left("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16032","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.16032/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.16032","created_at":"2026-07-20T02:19:27.349614+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.16032v1","created_at":"2026-07-20T02:19:27.349614+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.16032","created_at":"2026-07-20T02:19:27.349614+00:00"},{"alias_kind":"pith_short_12","alias_value":"GRKEOLS5NHUI","created_at":"2026-07-20T02:19:27.349614+00:00"},{"alias_kind":"pith_short_16","alias_value":"GRKEOLS5NHUI7PFL","created_at":"2026-07-20T02:19:27.349614+00:00"},{"alias_kind":"pith_short_8","alias_value":"GRKEOLS5","created_at":"2026-07-20T02:19:27.349614+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GRKEOLS5NHUI7PFLFUMPS5XZGA","json":"https://pith.science/pith/GRKEOLS5NHUI7PFLFUMPS5XZGA.json","graph_json":"https://pith.science/api/pith-number/GRKEOLS5NHUI7PFLFUMPS5XZGA/graph.json","events_json":"https://pith.science/api/pith-number/GRKEOLS5NHUI7PFLFUMPS5XZGA/events.json","paper":"https://pith.science/paper/GRKEOLS5"},"agent_actions":{"view_html":"https://pith.science/pith/GRKEOLS5NHUI7PFLFUMPS5XZGA","download_json":"https://pith.science/pith/GRKEOLS5NHUI7PFLFUMPS5XZGA.json","view_paper":"https://pith.science/paper/GRKEOLS5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.16032&json=true","fetch_graph":"https://pith.science/api/pith-number/GRKEOLS5NHUI7PFLFUMPS5XZGA/graph.json","fetch_events":"https://pith.science/api/pith-number/GRKEOLS5NHUI7PFLFUMPS5XZGA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GRKEOLS5NHUI7PFLFUMPS5XZGA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GRKEOLS5NHUI7PFLFUMPS5XZGA/action/storage_attestation","attest_author":"https://pith.science/pith/GRKEOLS5NHUI7PFLFUMPS5XZGA/action/author_attestation","sign_citation":"https://pith.science/pith/GRKEOLS5NHUI7PFLFUMPS5XZGA/action/citation_signature","submit_replication":"https://pith.science/pith/GRKEOLS5NHUI7PFLFUMPS5XZGA/action/replication_record"}},"created_at":"2026-07-20T02:19:27.349614+00:00","updated_at":"2026-07-20T02:19:27.349614+00:00"}